73.4. THE MAIN ESTIMATE 2471

= −⟨BX (t) ,X (t)⟩−⟨BX (s) ,X (s)⟩+2⟨BX (t) ,X (s)⟩+2⟨BX (t) ,X (t)⟩

−2⟨BX (s) ,X (t)⟩−2∫ t

s⟨Y (u) ,X (t)⟩du

= ⟨BX (t) ,X (t)⟩−⟨BX (s) ,X (s)⟩−2∫ t

s⟨Y (u) ,X (t)⟩du

Therefore,

⟨BX (t) ,X (t)⟩−⟨BX (s) ,X (s)⟩

= 2∫ t

s⟨Y (u) ,X (t)⟩du+ ⟨B(M (t)−M (s)) ,M (t)−M (s)⟩

−⟨BX (t)−BX (s)− (M (t)−M (s)) ,X (t)−X (s)− (M (t)−M (s))⟩

+2⟨BX (s) ,M (t)−M (s)⟩

The case with X0 is similar.The following phenomenal estimate holds and it is this estimate which is the main idea

in proving the Ito formula. The last assertion about continuity is like the well known resultthat if y ∈ Lp (0,T ;V ) and y′ ∈ Lp′ (0,T ;V ′) , then y is actually continuous a.e. with valuesin H, for V,H,V ′ a Gelfand triple. Later, this continuity result is strengthened further togive strong continuity. In all of this, X l

k and X rk are as described above, converging in K to

X .

Lemma 73.4.2 In the Situation 73.2.1, the following holds. For a.e. t

E (⟨BX (t) ,X (t)⟩)

< C(||Y ||K′ , ||X ||K , ||Z||J ,∥⟨BX0,X0⟩∥L1(Ω)

)< ∞. (73.4.8)

where K,K′ were defined earlier and

J = L2([0,T ]×Ω;L2

(Q1/2U ;W

))In fact,

E

(sup

t∈[0,T ]∑

i⟨BX (t) ,ei⟩2

)≤C

(||Y ||K′ , ||X ||K , ||Z||J ,∥⟨BX0,X0⟩∥L1(Ω)

)Also, C is a continuous function of its arguments, increasing in each one, and C (0,0,0,0)=0. Thus for a.e. ω,

supt /∈NC

ω

⟨BX (t,ω) ,X (t,ω)⟩ ≤C (ω)< ∞.

Also for ω off a set of measure zero described earlier, t→ BX (t)(ω) is weakly continuouswith values in W ′ on [0,T ] . Also t→ ⟨BX (t) ,X (t)⟩ is lower semicontinuous on NC

ω .